\(\phi ^{\mp }(x)\)  is the negative and positive field operator of the field and \(\chi ^a_{\mu }\)  are the
corresponding complex coefficients [1]}. The system + field formed a closed system.
To find a dynamical equation of the system, the starting point is the “Von Neumann-Liouville" equation
[2]}. It is given by,
\(\frac{d \rho (t)}{dt}&=& -i [\mathcal {H}_s+\mathcal {H}_f+\mathcal {H}_{sf}, \rho (t)]~.\) 
